2 ijcmp oct-2017-2-nuclear structure calculations

International journal of Chemistry, Mathematics and Physics (IJCMP) [Vol-1, Issue-3, Sep-Oct, 2017]
https://dx.doi.org/10.22161/ijcmp.1.3.2 ISSN: 2456-866X
www.aipublications.com/ijcmp Page | 5
Nuclear Structure Calculations using Different
Symmetries of 20
Ne nucleus
Azza O. El-Shal, N. A. Mansour*
, M. M. Taha, Omnia S. A. Qandil
Mathematics and Theoretical Physics Department, Nuclear Research Center, Atomic Energy Authority, Cairo, Egypt.
*
Physics Department, Faculty of Science, Zagazig University,
Zagazig, Egypt.
Abstract— Using the two forms of Fish-Bone potential (I and II), a self-consistent calculations are carried
out to perform the analysis of binding energies, root mean square radii and form factors using different
configuration symmetries of 20
Ne nucleus. A computer simulation search program has been introduced to
solve this problem. The Hilbert space was restricted to three and four dimensional variational function
space spanned by single spherical harmonic oscillator orbits. A comparison using Td and D3h configuration
symmetries are carried out.
Keywords— Alpha Clustering / Configuration Symmetry /Fish-Bone potential.
I. INTRODUCTION
The concept of cluster structure in nuclei has been a subject of interest since the early days of nuclear
physics till now [1]-[3]. The idea is largely supported by the fact that alpha particles have exceptional
stability and that they are the heaviest particles emitted in natural radioactivity. Various models have been
proposed to account for possible nuclear clustering and to study its effects [4]. Later, a simple version of the
model, in which the alphas are assumed to have no internal structure is considered. The concept of alpha-
clustering has found many applications to nuclear reactions and nuclear structures [5,6] .
Many such clusters are possible in principle, but the formation probability depends on the stability of the
cluster, and of all possible clusters the alpha particle is the most stable due to its high symmetry and binding
energy. Thus a discussion of clustering in nuclei is mainly confined to alpha-particle clustering. For the α-
nuclei 8
Be, 12
C, 16
O, 20
Ne, 24
Mg, 28
Si, 32
S, the geometrical equilibrium configuration of the α-particles are
generally assumed to belong to the symmetry point groups D∞h, D3h, Td, D3h, Oh (or alternatively D4h) ,
D5h, D6h, respectively. The 20
Ne have also two other configuration symmetries; the Td configuration
symmetry and the D2d configuration symmetry in addition to the D3h configuration symmetry.
In each configuration studied, it was assumed that the nucleons form persistent alpha-particle clusters
arranged in some symmetric fashion.
The Fish-Bone potential [7] of composite particles simulates the Pauli effect by nonlocal terms. The α-α
fishbone potential be determined by simultaneously fitting to two-α resonance energies, experimental phase
shifts, and three-α binding energies. It was found that, essentially, a simple Gaussian can provide a good
description of two-α and three-α experimental data without invoking three-body potentials. Many authors
adopted the fishbone model because, in their opinion, this is the most elaborated phenomenological cluster-
model-motivated potential. The variant of the fishbone potential has been designed to minimize and to

neglect the three-body potential. Therefore, they can try to determine the interaction by a simultaneous fit to
two- and three-body data [8].
The aim of this work is to employ the D3h configuration symmetry and the Td configuration symmetry of
20
Ne nucleus and using the Fish-Bone potential of type I and II to obtain the binding energies, root mean
square radius, and the form factors of the 20
Ne nucleus.
II. THE THEORY
We consider a system of N identical bosons described by a Hamiltonian of the usual form
Where t(i) is the kinetic energy operator ith
particle and v(i,j) the two-body interaction. In Hartree-Fock
method, one takes for the best choice of the normalized wave function  the one that it minimizes the
expectation value of the Hamiltonian H
In most Hatree-Fock calculations for light nuclei one has taken the subspace spanned by the lowest
harmonic oscillator shell l j >. We assume that all the particles occupy the same orbital  belonging to the
average field. Hence the intrinsic state of the whole system would be described by the symmetric wave
function
In this subspace, the HF orbitals l > are then determined by their expansion coefficients m
j .
And the HF-Hamiltonian h(1 , 2 ,….. N) is replaced by the matrix
Where
)1(),()(ˆˆ
1 1
  

N
i
N
ji
jivitVTH
)3()().........3()2()1(),.....,2,1( NN  
)4(
1


N
j
j jm

)5(


ls
k l
k mljvkimjtijhi  

)6(jlvkiljvkiljvki s

)2(0 H

The HF equations are then replaced by the matrix equation
One proceeds by iteration until self-consistency is achieved
Fish-Bone potential :
A fishbone potential of the α-α system was determined by Schmid E. W. [9] . The harmonic oscillator
width parameter was fixed to a = 0.55 fm−2
and the local potential was taken in the form
)r
a
(erf
r
e
)rexp(v)r(Vl
3
24 2
2
0   (8)
where v0 = -108.41998MeV and β = 0.18898fm-2
and called Fish-Bone I(FB-1)While this potential provides
a reasonably good fit to l = 0 and l = 2 and l = 4 partial wave phase shifts, it seriously overbinds the three-α
system as shown in ( table 1) . The experimental binding energy of the three-α (12
C) system is E3α = -7.275
MeV.
Table 1
L = 0 three- binding energy as a function of subsystem angular momentum lmax in case if fish-bone
potential of Kircher and Schmid (FB-1)[9] and the results of Papp Z. and Moszkowski S. (FB-2).
Lmax FB-1 FB-2
2 0.057 -0.313
4 -15.47 -7.112
6 -15.63 -7.273
8 -15.63 -7.275
One may conclude that there is a need for three-body potential. This was the choice Oryu and Kamada[10].
They added a phenomenological three-body potential to the fish-bone potential of Kircher and Schmid[9]
and found that a huge three-body potential is needed to reproduce the experimental data. But, Faddeev
calculations[8] reveal that the  = 4 partial wave is very important to the three- α binding and, for this
partial wave, the fit to experimental data is not so stellar. So Papp Z.and Moszkowski S.[8] concluded, that
it may be possible to improve the agreement in the  = 4 partial wave and achieve a better description for
the three- α binding energy in the point view of the these authors.
Thus as a local potential, two Gaussians plus screened Coulomb potential are added to form Fish-
Bone II(FB-2):
)7( 
j
ij mmjhi 


)
3
2
(
4
)exp()exp()(
2
2
22
2
11 r
a
erf
r
e
rvrvrVl   (9)
where v1, β1, v2, β2 and a are fitting parameters. In the fitting procedure Papp Z. and Moszkowski S.
incorporated the famous 8
Be, l = 0 resonance state at exp
2E b = (0.0916 -0.000003i) MeV, the 12
C three- α
ground state energy exp
3E b = -7.275 MeV, and the l = 0, l = 2 and l = 4 low energy phase shifts. With
parameters v1 = -120.30683493 MeV, β1 = 0.20206127 fm-2
, v2 = 49.06187648 MeV, β2 = 0.76601097 fm-2
and a = 0.64874009 fm-2
, they achieved a perfect fit.
For the l = 0 two-body resonance state they get E2b = 0.09161 – 0.00000303i MeV, and for the three-body
ground state E3b = -7.27502 MeV. Notice that unlike with the Ali-Bodmer potential, they achieved this
agreement by using the same potential in all partial waves. Having this new α-α fishbone potential from
the fitting procedure, they also calculated the first excited state of the three- system. This state is a resonant
state, and we got 3E res
 = (0.54- 0.0005i) MeV, which is again very close to the experimental value.
Solutions in a Three and Four Dimensional Space :
We consider solutions in a three dimensional variational space spanned by the orthonormal states l 1> , l
2 > and l 3> . In this case, we have twenty-one different symmetrized two-body matrix elements.
A HF-orbital  will have the general form
Where
And assume that the coefficient mj ‘s are real.
To investigate the HF-solutions , we have to specify the alpha-alpha potential. The specific combinations
chosen as the basic states depend on the symmetry of the intrinsic structure that is expected from the
molecular alpha-particle model.
Now we chose basic states which are invariant with respect to the transformation of the symmetry group
Td [11]. Therefore, we chose our three basic states as
3
1
(10)j
j
m j


 
, (11)j j j j jj
j
m m m m   


  
    

001 s (12)
 2020
2
1
2  ff (13)
 40400
2
1
3
2


 gogg

(14)
Where  is a parameter determined from the Td symmetry.
We chose also basic states which are invariant with respect to the transformation of the symmetry group
D3h . Therefore, we chose our four basic states as
001 s (15)
 0001
1
1
2
2
dcs
c


 (16)
 330
2
1
3  off (17)
004 g (18)
Here the oscillator shell model wave functions are given by
),()(, lmnl YrRnlmr 

(19)
)/rexp()/r(L
)/r()]/ln(/)n([)r(R
/l
n
l/
nl
2
0
22
0
221
0
213
0
2
2312





(20)
0=(h/m)1/2
is the oscillator parameter.

Nuclear Density and Form Factor :
For a system of number of alpha particles in the above basic states, the corresponding charge density
normalized to unity, is readily found to be

lm
lmlm ),(Y)r()r(  (21)
The form factor corresponding to the spherical part of the charge density can be readily evaluated, one
readily obtains
drerqF rqi

.
)()(   (22)
where q is the momentum transfer. The expression of F(q) has to be multiplied by a form factor Fα which
account for the -particle distribution where
22


q
e)q(F 
 (23)
Fish-Bone Potential I Calculations :
Using the Fish-Bone potential I (FB-1) according to its equation (8), we have the following results of the
binding energies and root mean square radii of 20
Ne nuclei, where the alpha particles are arranged according
to the Td and D3h symmetries.
Table 2
The parameters used in our calculations of fishbone
Potential I of 20
Ne
Symmetry V0(MeV) β1(fm-2
) a (fm-2
)
Td -9 0.1889 0.55
D3h -19 0.1889 0.55

Table.3
The calculated binding energies and root mean square radii of 20
Ne using the Fish-Bone potential I
Symmetry Theor. B.E.(MeV) Exp.B.E.(MeV) Theor. rms (fm) Exp. Rms (fm)(a)
Td -19.397 -19.18 2.711 2.91±0.05
D3h -20.454 -19.18 2.833 2.91±0.05
Fish-Bone Potential II Calculations :
Using the Fish-Bone potential II (FB-1) as in equation(9), we have the following results of the binding
energies, root mean square radii and form factors
Table.4
The parameters used in our calculations of Fish-Bone Potential II of 20
Ne
Table.5
The calculated binding energies and root mean square radii of 20
Ne using the Fish-Bone potential II
Symmetry Theor.
B.E.(MeV)
Exp.B.E.(MeV) Theor. rms (fm) Exp. Rms (fm)
Td -23.385 -19.18 2.963 2.91±0.05
D3h -20.368 -19.18 2.847 2.91±0.05
Symmetry V1(MeV) V2(MeV) β1(fm-2
) β2(fm-2
) a (fm-2
)
Td -97 39 0.2021 0.7660 0.6487
D3h -65 21 0.2021 0.7660 0.6487

0 1 2 3 4
1E-11
1E-10
1E-9
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
1
IFch
(q)I2
q(fm
-1
)
other theo.[13]
Exp. [12]
Theo.(present work)
Fig. 1 : The form factor of 20
Ne which have the configuration symmetry Td and using
the parameters of Table 4 of FB-II .The points represent experimental values.
0 1 2 3 4
1E-11
1E-10
1E-9
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
1
IFch
(q)I2
q(fm
-1
)
other theo. [13]
Exp. [12]
Theo.(present work)
Fig. 2: The form factor of 20
Ne which have the configuration symmetry D3h and using
the parameters of Table 2 of FB-I. The points represent the experimental values.

0 1 2 3 4
1E-11
1E-10
1E-9
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
1
1Fch
(q)12
q(fm
-1
)
other theo.[13]
Exp.[12]
theo. (present work)
Fig. 3: The form factor of 20
Ne which have the configuration symmetry D3h and using
the parameters of Table 4 of FB-II. The points represent the experimental values.
III. DISCUSSION
By using the two types of the Fish-Bone potentials I and II and with the oscillator parameter Bt for 20
Ne in
addition to the parameter of the alpha particle Bα ,and also considering the different configuration
symmetries of 20
Ne; Td and D3h symmetries, we have got the nuclear structure properties of the 20
Ne nuclei
such as binding energies, root mean square radii and form factors, and comparing the results with the
experimental elastic scattering charge form factor [12] and other theoretical ones[13].
In case of Fish –Bone potential I (FB-1) , the binding energy of 20
Ne nuclei and the root mean square radius
are in good agreement with the experimental results as shown in Table (3), but we have not got form factors
for 20
Ne nuclei in case of considering that the 20
Ne nucleus posses the configuration symmetry Td . On the
other hand, when we consider the four dimensional functional space, we have got the form factor as shown
in Fig. (1). Applying the Fish-Bone potential II, we have got the binding energy, the root mean square
radius and the form factors for 20
Ne nuclei either we consider the 20
Ne nuclei have Td symmetry or D3h
symmetry as shown in Figs.(2) and Fig. (3) .
We conclude that the configuration symmetry D3h of 20
Ne nucleus yielded better results than that of Td
symmetry, and the results are in acceptable agreement with the experimental points in addition to other
theoretical ones.

References
[1] Ripka G. Baranger M. and Vogt E. (eds.) , Advances in Nuclear Physics vol. 1, Phenum (1968).
Neelam Guleria, Shashi K. Dhiman,Radhey Shyam, Nucl. Phys. A886 (2012)71.
Kazuyuki Sekizawa, Kazuhro Yabana, Arxiv: 1303.0552 [ nucl-th], (2013).
[2] Ripka G. and Porneuf M. (eds.), Proceedings of the International Conference on Nuclear Self –
Consistent Fields,North Holland, (1976).
Gnezdilov N. V., Borzov I. N., Saperstein E. E. and Tolokonnikov S. V. Arxiv :1401.1319v4 [nucl-
th],(2014).
[3] Moszkowski S. A. ' Encyclopedia of Physics ' Springer Verlag, Heidebberge, (1957).
[4] Brink D. M. ' Proceedings of International School of Physics ' Enerco Fermi , Acadmeic Press, New
York (1966).
[5] Kramer P. , Clustering Phenomena in Nuclei, Proceedings of the International Symposium, Attemp
Verlag, Tubingen (1981), Zbigniew Sosin et al., Eur. Phys. J. A, 52: 120(2016) .
[6] Bech C. ' Clustering Effect Induced by Light Nuclei' ,Proceeding of the 12th
Cluster Conference,
Debrecen, Hungary –Septemper (2012).
[7] Day J. P., McEwen J. E., Elhanafy M., Smith E., Woodhouse R., and Papp Z., Phy. Rev. C 84,
034001 (2011), Arkan R. Ridha ,Iraqi Journal of Physics, Vol.14, No.30, PP. 42-50(2016).
[8] Papp Z. and Moszkowski S., Arxiv: 0803.0184v2 [nucl-th] ,(2008).
[9] Schmid E. W. ,Z. Phys. A297 (1980)105, Schmid E. W. , Z. Phys. A397 (1981) 311.
Kircher R. and Schmid E. W., Z. Phys. A 299, 241 (1981) .
[10] Oryu S. and Kamada H., Nucl. Phys. A493 (1989)91.
[11] El-Shal Azza O.,Mansour N. A.,Taha M. M. and Qandil S. A.,Journal of Applied Mathematics and
Physics, 2(2014),869.
[12] Ajzenberg F.-Selove, Nucl. Phys. A 300 (1978) 148.
[13] E.V. Inopin et al., Annals of physics 118, 307-340 (1979).

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